Beam divergences of high-order extreme ultraviolet harmonics from intense laser interactions with steep plasma density gradients are studied through experiment and Fourier analysis of the harmonic spatial phase. We show that while emission due to the relativistically oscillating mirror mechanism can be explained by ponderomotive surface denting, in agreement with previous results, the divergence of the emission due to the coherent wake emission mechanism requires a combination of the dent phase and an intrinsic emission phase. The temporal dependence of the divergences for both mechanisms is highlighted while it is also shown that the coherent wake emission divergence can be small in circumstances where the phase terms compensate each other.